The field of an infinite charged line decreases as \(1 / r .\) Why isn't this a violation of the inverse-square law?

Understanding the electric field strength is integral to grasping the underpinnings of electrostatics. The electric field strength, often denoted as E, is a vector quantity that measures the force per unit charge experienced by a small positive test charge placed within the field. Mathematically, it is expressed as \( E = F / q \), where E is the electric field strength, F is the force experienced, and q is the charge. In an intuitive sense, the electric field strength represents the 'push' or 'pull' that a charge would feel in the presence of another charge. A larger field strength means a stronger force on a test charge, and in visual terms, closely spaced field lines on a diagram indicate this higher strength.
Because of its vector nature, the direction of the electric field strength also tells us the direction in which a positive test charge would move if placed within the field. This direction is always radially away from positive charges and towards negative charges. It's crucial to note though, the electric field strength decreases as you move farther away from the source charge, and the rate at which it decreases depends on the geometry of the charge distribution, as illustrated by the inverse-square law for point charges and the linear decrease with distance for continuous charge distributions such as a charged line.

A point charge is a hypothetical charge located at a single point in space with no volume. Despite being an idealization (since actual charges occupy some space), the concept is pivotal in electrostatics, leading to valuable approximations and understanding of reality. The electric field created by a point charge is symmetrical and emanates uniformly in all directions, conforming to the inverse-square law. This law dictates that the field strength decreases with the square of the distance from the point charge.
For a point charge Q at a distance r, the electric field strength E can be calculated using Coulomb's law, expressed as \( E = k \frac{|Q|}{r^2} \), where k is Coulomb's constant. This relationship showcases that for point charges, as the distance doubles, the field strength becomes one-fourth, illustrating the essence of the inverse-square law. The simplicity of this model makes it a fundamental concept for understanding the forces between charges in a vacuum and is often a starting point in electric field analysis.

A charged line is another key concept in electrostatics, concerning a line that is uniformly charged along its length. Unlike a point charge, a charged line extends infinitely in one dimension, creating a field that diminishes differently with distance. For a charged line, our intuition and the inverse-square law are reshaped due to the line's unique geometry. The electric field vectors extend radially outwards in all directions perpendicular to the line, forming cylindrical surfaces as equipotential areas around the line.
In an infinitely long charged line, the electric field strength decreases linearly with the distance (1/r) because the expansion of field lines occurs over a two-dimensional plane rather than a three-dimensional space. This results from the constant linear charge density along the line, in contrast to a point source where the density spreads in all directions. We can understand this by imagining the field lines emanating from a small segment of the line and observing how their density decreases over distance. The ‘spread’ of the field lines isn't as dramatic as that from a point charge, leading to a less rapid decrease in field strength. This concept underscores the importance of the shape and dimensionality of charged objects when determining how electric fields propagate through space.